Calculate the flux of the vector field F(x, y, z) (5x + 8)i through a disk of radius 7 centered at the origin in the yz-plane, oriented in the negative x- direction.
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- consider the parallelepiped with sides: A=3i+2j+k، B=i+j+2k, c=i+3j+3k, then 1-Find the rolume of the paralldepiped 2-Find the area of the face determined by A and B. 3-Find the angle between the vactor C and the plane containing the face determined by A and BCalculate the flux of the given vector field by evaluating the line integral directly alongthe given curve for the below parts:(a) The vector field is ⃗ F = (x − y)⃗i + x⃗j. The curve is the circle x^2 + y^2 = 1in the xy-plane. Use the parameterization x = cos t and y = sin t.(b) The vector field is ⃗ F = (x − 1)⃗i + y⃗j. The curve is a circle of radius 3centered at (1, 1). The parametric form of this circle is⃗r = (1 + 3 cos t)⃗i + (1 + 3 sin t)⃗j, 0 ≤ t ≤ 2π(c) The vector field is ⃗F = x⃗i + y⃗j. The curve is the line segment from thepoint (0, 1) to the point (1, 3).The electric field in a region of space near the origin is given by E(z, y, :) – E, (*) yî+ xî a (a) Evaluate the curl Vx E(x, y, z) (b) Setting V(0, 0, 0) = 0, select a path from (0,0, 0) to (x, y, 0) and compute V (r, y,0). (c) Sketch the four distinct equipotential lines that pass through the four points (a, a), (-a, a), (-a, -a), and (a, -a). Label each line by the value of V.
- Find the flux of F = xi - 2yj + zk across the portion of cylinder x² + z² = 9 in the first and forth octants. (3,-3,0) n X (0,0,3) (3,0,0) (0,3,0) yGiven a non-uniform SURFACE charge density o= ko r z cos q, ko constant, r, o,z are cylindrical coordinate variables 12. Find the total "Q" placed on a closed cylinder of radius "R" and length "L" centered on the origin with one base in the x-y plane, along the positive z-axis.Compute the flux of the vector field F = 2zk through S, the upper hemisphere of radius 5 centered at the origin, oriented outward. flux =